One of the many equivalent formulation Köthe's conjecture is assertion that there exists no unital ring which contains two nil right ideals whose sum not nil. We discuss several consequences an observation if Koethe fails then a counterexample in form countable local subring suitable self-injective prime von Neumann regular ring.

We presented a simple and direct way to construct unipotent unit clean but not nil-clean element in ring. New examples of unipotent/unit-regular elements that are given. also study the product two idempotents/unit-regulars which unit-regular. The studies exemplified subrings M_2 (Z).

The main result of this paper is a generalization the theorem Chevalley-Shephard-Todd to rings invariants pseudoreflection groups over Dedekind domains. In special case principal ideal domain in which group order invertible it proved that ring isomorphic polynomial ring. An intermediate every finitely generated regular graded algebra tensor product blowup algebras.

Several characterizations are given of (Zelmanowitz) regular modules among the torsionless modules, the locally projective modules, the nonsingular modules, and modules where certain submodules are pure. Along the way, a version of the unimodular row lemma for torsionless modules is given, and it is shown that a regular ring is left self-injective if and only if every nonsingular left module is...

In this paper the arithmetic Chow groups and their product structure are extended from the category of regular arithmetic varieties to regular Deligne-Mumford stacks proper over a general arithmetic ring. The method used also gives another construction of the product on the usual Chow groups of a regular Deligne-Mumford stack.

We prove the simple fact that the factor ring of a Koszul algebra by a regular, normal, quadratic element is a Koszul algebra. This fact leads to a new construction of quadratic Artin-Schelter regular algebras. This construction generalizes the construction of Artin-Schelter regular Clifford algebras. 1991 Mathematics Subject Classification. 16W50, 14A22.

The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety’s cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer’s work more transparent and accessible by presenting the cohomology ring as a graded quotient of a poly...

On minimal completely regular spaces associated with a given ring of continuous functions." Michigan

For a frame $L$, consider the $f$-ring $ mathcal{F}_{mathcal P}L=Frm(mathcal{P}(mathbb R), L)$. In this paper, first we show that each minimal ideal of $ mathcal{F}_{mathcal P}L$ is a principal ideal generated by $f_a$, where $a$ is an atom of $L$. Then we show that if $L$ is an $mathcal{F}_{mathcal P}$-completely regular frame, then the socle of $ mathcal{F}_{mathcal P}L$ consists of those $f$...

We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author [Loc07]. We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived catego...